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Mathematics > Differential Geometry

Title:
A generalization of manifolds with corners

Abstract: In conventional Differential Geometry one studies manifolds, locally modelled
on ${\mathbb R}^n$, manifolds with boundary, locally modelled on
$[0,\infty)\times{\mathbb R}^{n-1}$, and manifolds with corners, locally
modelled on $[0,\infty)^k\times{\mathbb R}^{n-k}$. They form categories ${\bf
Man}\subset{\bf Man^b}\subset{\bf Man^c}$. Manifolds with corners $X$ have
boundaries $\partial X$, also manifolds with corners, with $\mathop{\rm
dim}\partial X=\mathop{\rm dim} X-1$.
We introduce a new notion of 'manifolds with generalized corners', or
'manifolds with g-corners', extending manifolds with corners, which form a
category $\bf Man^{gc}$ with ${\bf Man}\subset{\bf Man^b}\subset{\bf
Man^c}\subset{\bf Man^{gc}}$. Manifolds with g-corners are locally modelled on
$X_P=\mathop{\rm Hom}_{\bf Mon}(P,[0,\infty))$ for $P$ a weakly toric monoid,
where $X_P\cong[0,\infty)^k\times{\mathbb R}^{n-k}$ for $P={\mathbb
N}^k\times{\mathbb Z}^{n-k}$.
Most differential geometry of manifolds with corners extends nicely to
manifolds with g-corners, including well-behaved boundaries $\partial X$. In
some ways manifolds with g-corners have better properties than manifolds with
corners; in particular, transverse fibre products in $\bf Man^{gc}$ exist under
much weaker conditions than in $\bf Man^c$.
This paper was motivated by future applications in symplectic geometry, in
which some moduli spaces of $J$-holomorphic curves can be manifolds or
Kuranishi spaces with g-corners (see the author arXiv:1409.6908) rather than
ordinary corners.
Our manifolds with g-corners are related to the 'interior binomial varieties'
of Kottke and Melrose in arXiv:1107.3320 (see also Kottke arXiv:1509.03874),
and to the 'positive log differentiable spaces' of Gillam and Molcho in
arXiv:1507.06752.

Comments:

97 pages, LaTeX. (v3) final version, to appear in Advances in Mathematics